We have seen where the word Science comes from: scire means to cut, split (as in scissors), separate, decide true from false. We, like other living creatures, do so on the basis of evidence – what we see there is. We use evidence to update our beliefs. We are all Bayesian.

Despite Kant’s grand attempt to salvage some of it, there is no such thing as a priori knowledge. What may appear to us as transcendent knowledge, emanating from pure reason independent of evidence, is and can only be based on notions – concepts, principles, axioms – that we regard as self-evident.

Such notions are the subject of Metaphysics. The word came about, apparently, to denote the collection of Aristotle’s treaties that his late editors arranged to place after (meta) his Physics. Whereas Aristotle himself had not called them Metaphysics, actually referring to them as ‘first philosophy’, dealing with concepts that came before Physics in importance and generality.

Be as it may, we can think of metaphysics as the area we enter once we start running out of answers to our Why questions. Answers are local explanations built on our own hard evidence or, most often, on soft evidence emanating from trusted sources. We learn to accept local explanations and live with them, but every answer begets new questions, in a seemingly endless why-chain whose infinity we find impossible to accept. Explanations cannot go on forever. At some point, even the cleverest dad succumbs to the urge to end his child’s relentless barrage of whys with a resounding last answer: ‘because that’s the way it is!’

But to the undaunted child dad’s last answer turns into the ultimate question: What is the way it is? Once we set out to answer this question we have entered the land of metaphysics. Metaphysics is mankind’s effort to establish the absolute, unquestionable and irrefutable episteme that stands firm above Physics. Episteme is knowledge that does not need evidence because it is self-evident, certain without experiment and secure from the perils of experience.

How can we achieve such knowledge? Clearly, we can’t reach it from the side of experience, whence we can only expect an infinite regress of explanations. So it must come from the other side. But what’s on the other side? Clearly, we know nothing about it – if we did, we would have already gone past the answer we are looking for. As Immanuel Kant put it, noumena are on the other side – things-in-themselves, absolutely unknowable and irremediably inaccessible to our mind. All we can know are phenomena – things as they appear to us in the light of evidence.

Metaphysics is the boundary between phenomena and noumena – a boundary that mankind would love to cross but can only push forward, unfolding and accumulating new and better physical explanations of natural phenomena. Such is the love at the root of philosophy – the ever-burning, insatiable desire for sophia, the supreme wisdom in whose full light we would finally be able to contemplate the way it is. But the light of philosophy is the same light that illuminates phenomena. Metaphysics is and can only be on the side of phenomena – the side of experience and evidence. In the words of Arthur Schopenhauer:

Metaphysics thus remains immanent, and does not become transcendent; for it never tears itself entirely from experience, but remains the mere interpretation and explanation thereof, as it never speaks of the thing-in-itself otherwise than in its relation to the phenomenon. (Will, Volume II, p. 183).

Metaphysics is not and cannot be a priori knowledge, independent of evidence. Its value does not rest on its being beyond evidence, but on being based on notions that we regard as self-evident. Like mathematics and geometry, metaphysics is an axiomatic system – true insofar as its axioms are true. An axiom is that which is thought worthy, weighty, and thus bears authority – a concept interestingly close to the original meaning of probability. Axioms are statements assumed to be self-evidently true, thus requiring no proof or demonstration. Given the axioms, the theorems built on them using truth-preserving rules of inference are demonstrably true.

As such, the validity of an axiomatic system depends on the weight of its axioms. The more precise, clear, obvious, intuitive, indubitable the axioms, the stronger the system. Take Euclid’s Elements, which, as we know, is built on five axioms (or postulates). As we have seen, one can argue about the fifth. But not about the first: A straight line can be drawn joining any two points. Or the second: A finite straight segment can be extended indefinitely into a straight line. The third: From any straight segment a circle can be drawn having the segment as radius and one endpoint as centre. And the fourth: all right angles are equal. A geometry in which any of these four axioms is untrue is even hard to imagine. They are glaringly, unquestionably self-evident.

Now let’s compare it to Spinoza’s Ethics, which he explicitly wrote along the lines of Euclid’s Elements.

Here is its first axiom: ‘Everything which exists, exists either in itself or in something else’. The second: ‘That which cannot be conceived through anything else must be conceived through itself’. And the third, which we have encountered as the Principle of Sufficient Reason: ‘From a given definite cause an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow’. And so on. One may or may not agree with any of these statements – provided that he truly understand what they mean. But it would be at least preposterous to regard them as self-evident.

And what about Definitions, which in Elements as well as in Ethics precede the Axioms? Let’s take the first three. In Elements they are: 1) ‘A point is that which has no part’. 2) ‘A line is breathless length’. 3) ‘The ends of lines are points’. Hard to disagree. But in Ethics: 1) ‘By that which is self-caused, I mean that of which the essence involves existence, or that of which the nature is only conceivable as existent’. 2) ‘A thing is called finite after its kind, when it can be limited by another thing of the same nature; for instance, a body is called finite because we always conceive another greater body. So, also, a thought is limited by another thought, but a body is not limited by thought, nor a thought by body’. 3) (we have seen this one) ‘By substance, I mean that which is in itself, and is conceived through itself: in other words, that of which a conception can be formed independently of any other conception’.

Whaaat? Definitions and axioms can only be as clear as the terms that compose them. We all know and agree on what a point, a straight line and a circle are. But what about essence and existence, cause and substance? They are much more complex, vaguer and harder concepts to define and comprehend. It’s no wonder, then, that all the ensuing Propositions in Ethics are, let’s say, less cogent than Pythagoras’s theorem. Take, for instance, Proposition XI, Part I:

God, or substance, consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists.

Here is the proof:

If this be denied, conceive, if possible, that God does not exist: then his essence does not involve existence. But this (Prop. VII) is absurd. Therefore God necessarily exists. Q.E.D.

Uhm. And what is Proposition VII?

Existence belongs to the nature of substances.

and its proof:

Substance cannot be produced by anything external (Corollary, Prop. VI), it must, therefore, be its own cause – that is, its essence necessarily involves existence, or existence belongs to its nature. Q.E.D.

Oh well. I spare you Proposition VI and its Corollary. Spinoza was a great philosopher and an admirable man, and his Ethics is a trove of powerful thoughts and ideas. But its metaphysical value can only be as compelling as its murky foundations.

This is metaphysics’ typical pitfall. While usually conceived as the product of pure reason, standing above physics and unrestrained by experience, metaphysics can’t really be nothing else than a more or less coherent inferential system which is in fact so entwined with evidence as to be entirely based on supposedly self-evident foundations.

The trouble is that self-evidence is in the eye of the beholder. And – as we have seen repeatedly throughout this blog – it is amazing what different people, from the dimmest to the supremely intelligent, come to regard as self-evident. Once one is satisfied that he has made all the way through why-chains to answering the ultimate question, and that he finally knows the way it is, it is tempting to invert direction and reinterpret reality in the light of his newfound metaphysical principles.

I had never spent much time thinking about Bitcoin. After reading a couple of articles to figure out what it was, I associated it with the muddled assemblage of Austrian devotees ranting against central banking, fiat currency, ‘big government’, ‘the elites’ and ‘the establishment’, and left it at that.

But then the other day, when my basketball buddy Adam, who is trading cryptocurrencies, asked me what I thought about them, I realised I needed a proper answer. Fighting my flippancy impulse – last year I had lashed out at Ed on Brexit and at John on Hillary Clinton ‘corruptness’ (Bernie Sanders’ flavour, not Trump’s) – I just told Adam that I hadn’t given it much thought. But that was not acceptable either. I had to have a closer look.

Luckily (HT @manualofideas) I soon found this recent post on Aswath Damodaran’s blog, which, in typical crystal clarity, makes all the relevant points. The post, written with Bitcoin at $6,100, should be read alongside an earlier post, written only a few months ago, when Bitcoin was priced at $2,800, and a later post, written a few days ago in response to critics. In a nutshell:

Bitcoin is not an asset, because it does not generate future cash flows. As such, it does not have a value.

Bitcoin is a currency, enabling the exchange of goods and services. As such, it has a price, relative to other currencies.

The relative price of a currency depends on its quality as a unit of account, a medium of exchange and a store of value.

One can invest in assets, based on an estimation of their intrinsic value, but can only trade in currencies, based on the anticipation of their future price movements. Buying Bitcoin is not an investment.

What I didn’t know is how many cryptocurrencies there are beside Bitcoin: 1221 of them at the last count – with fancy names like Ripple, IOTA, Qtum, Stellar Lumens – for a total market cap of $169 billion! Each has its own website, detailing how different and better they are compared to the others, and each can be traded on dozens ‘exchanges’ – with other fancy names like Bithumb, Coinone, YoBit, Poloniex. Most of them have explosive price charts, and Adam feels very good about it – he’s been buying more beer rounds. But what will be the dollar price of IOTA a year from now? Like Damodaran, I am not saying it will be zero. But I can’t see how anybody could have any idea.

I will let Adam ponder upon Damodaran’s analysis. As an addition to his considerations, I see his table contrasting the Pricing Game and the Value Game as a striking illustration of the ruinous influence of the Efficient Market Theory.

By collapsing Value into Price, the EMT turns an honourable intellectual pursuit into a vacuous guessing game, where thinking is overruled by action, patience by speed and brains by guts. If prices are always where they should be, and only new information can change them, then success is determined by how quickly one is able to collect and react to news. High-frequency, algorithmic and other types of ‘quant’ trading are a direct offspring of the EMT. And so is home-made online trading, as well as its mirror image, index funds. They all make a mockery of the noble art of investing.

On the way back to London from Italy earlier this month, I decided to stop in Basel. It was mid-way and it had long been on the list of cities I wanted to visit. Why it was on that list started to surface as I picked a hotel on Trivago. Euler Hotel – definitely. We arrived in the evening and the boys were keen to get back home. So I only had half a day the following morning.

Basel’s old town centre is quite small and its main landmark is the Münster, a Romanesque church with a long and interesting history. As we waited for its doors to open at 10, I started touring the adjacent cloister. One of the highlights of the place is that Erasmus was buried there in 1536 – a sudden death following an attack of dysentery. But while looking for the grave in the cloister, wandering among tombs and commemorative plates of the city’s notables, one of them gave me a jolt:

Jacob Bernoulli, of course. He was born and lived in Basel his whole life, and died there on 16 August 1705 – morbo chronico, mente ad extremum integra – at the age of 50 years and 7 months.

Jacob – the eldest scion of the prodigious Bernoulli family – is one of my heroes. The author of the greatest masterwork in early probability theory, Ars Conjectandi, he is also credited as the first to discover the relationship between continuous compound interest and Euler’s number e, the base of natural logarithms. There – I suddenly realised – was a big piece of my subconscious attraction to Basel. Enchanted by my discovery, I asked my second child to pose for a photo next to the tombstone – my elder son was wandering somewhere else, supremely bored and impatiently waiting for lunch and departure.

After leaving the cloister, unable to come up with anything intelligible to say about Bernoulli, I told the kids about Erasmus and Paracelsus – another illustrious Basler. At 10 we visited the church – Erasmus’s grave is inside – and shortly after I realised my time was up – the children would have killed me if I had proposed any more ‘history stuff’. So we walked back to the Euler Hotel – Leonhard Euler was born in Basel two years after Jacob Bernoulli’s death. He was the first to use the letter e for the base of natural logarithms, apparently as the first letter of ‘exponential’, rather than of ‘Euler’. He also established the notation for π and for the imaginary number i, all beautifully joined together in Euler’s identity eiπ+1=0.

On the road to London, I kept thinking with delight at my semi-serendipitous encounter with Bernoulli. Then it struck me: I had seen that tombstone before. Back home, I checked. I was right: it was in one of the best books I have ever read, Eli Maor’s e: The Story of a Number.

As I reopened the book, it all came back to me: the Spira Mirabilis.

The logarithmic spiral is the curve r=aebθ in polar coordinates (r is the radius from the origin, θ is the angle between the radius and the horizontal axis, and a and b are parametric constants). Bernoulli had a lifelong fascination with the self-similar properties of the spiral:

But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might be not inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent, simillima filia matri. Or, if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light of light, remains ὁμοούσιος [consubstantial] with him, howsoever overshadowed. Or, if you prefer, since our spira mirabilis remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a symbol, either of fortitude and constancy in adversity, or, of the human body; which after all its changes, even after death, will be restored to its exact and perfect self; so that, indeed, if the fashion of imitating Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, Though changed, I rise again exactly the same, Eadem numero mutata resurgo.

This is the full quote from a paper by Reverend Thomas Hill published in 1875 (p. 516-517), from which Maor’s book takes an extract (p. 126-127), taken in turn from another book. Hill did not quote the source, but the original Latin quote can be found here (p. 185-186, available here), with the indication that it comes from a paper published by Bernoulli in the Leipsic Acts in 1692, which should be found here.

Bernoulli’s enthusiasm is easy to understand and to share. The logarithmic spiral is found in nature and art. The Golden spiral, whose growth factor b is the golden ratio, is a special case, and so is the circle as b tends to 0.

At the same time, it is difficult not to laugh at the manner in which Bernoulli’s wish was finally granted. Perhaps confused by the reference to Archimedes, the appointed mason cut an Archimedean spiral at the bottom of the tombstone, which has none of the properties Bernoulli so admired in the logarithmic spiral. And, to add insult to injury, he missed the word ‘numero’ from the motto. Bloody builders – always the same…

Bernoulli’s considerations made an impression on me when I first read Maor’s book. The spira mirabilis as a symbol of fortitude and constancy in adversity, or of the human body restored to its perfect self even after death. But after reading the passage in its entirety, I find it even more beautiful and inspiring. And how about taking a picture of my son – simillimus filius patri – next to the spiral, before any of this had come back to my mind?

By the way, my son’s name is Maurits, like (but not named after) M. C. Escher.

When I arrived at the Drayton Arms, he was already there. He had contacted me a few days earlier and we had arranged to meet for a drink. He worked for a head hunting firm, focused only – he was keen to specify – on investment management. After the introductory chit-chat, I made it clear that I was not interested in a job offer, and he made it clear that his purpose was to present his services to my firm’s potential hiring needs. With that out of the way, the conversation moved on amiably, flowing from market conditions to value investing, Brexit and other world affairs.

Until at one point – I can’t remember how and why – we veered towards terrorism, and from there to 9/11. “Of course” – said Sandeep, with the casual air of someone who is sharing the obvious among world-savvy, knowledgeable people “it was clearly an inside job”.

“What? What do you mean?” – I looked at him straight in the eyes.

“What? You don’t think so?” – Sandeep was genuinely taken aback by my sudden change of tone. Which, I agree, requires some explaining.

I have a Spinozan tolerance for freedom of opinion. It is the essence of Bayes: different priors, different information, or different interpretations of the same information, can give rise to different conclusions. This is obvious, and there is nothing wrong with it. But of course it doesn’t mean that anything goes. It means that, even when I have a strong view, I hold on to Cromwell’s rule and remain open to the possibility that, however high in my mind is the probability that I am right, I may be mistaken. As we know, hypothesis testing is the result of a tug of war between confirmative and disconfirmative evidence, which accumulates multiplicatively, leaving the possibility that, however overwhelming the evidence may be on one side, it may be annihilated by even one piece of conclusive evidence on the other. Another consequence of this framework is that, while I strive for certainty, I am comfortable with uncertainty: if neither side is strong enough to win the tug of war, there is nothing wrong with accepting that a hypothesis is only probably right, and therefore also probably wrong.

It is important to remember, however, that this only works insofar as one makes sure that evidence accumulation is as thorough as possible on both sides. This is easy to understand: there is no point gathering a lot evidence on one side while neglecting to do it on the other. One side will win nothing but a rigged game. But it is far from easy to do it in practice, as it requires fighting our natural tendency to succumb to the Confirmation Bias. The easier one side seems to be winning, the stronger should be our urge to reinforce the other side. It is by winning an ever tougher tug of war that we can aim to approach certainty.

This is an aptitude I have learned to nurture. The more I am convinced about something, the more I like to explore the other side, trying to distil its best arguments. If this succeeds in lowering my confidence, so be it: I feel richer, not poorer. And if it doesn’t, I am richer anyway, as I have built a clearer picture of what the other side stands on. This, after all, is what understanding means – distinct from justifying and, more so, from agreeing. The better one understands an argument, the easier it becomes to dismantle it and, perhaps, convince people on the other side to change their mind.

This is where sometimes I fail to keep my composure: when I face a conviction based on a pile of one-sided arguments, typically soaked in hyperbolic language, which blatantly misrepresents, disregards or belittles the other side. But what really gets on my nerves is a dirtier trick: when the balance of evidence is overtly on one side, the only way to overturn the verdict is to find – or, failing that, make up – a conclusive piece of evidence on the other side. This is the standard trick employed by conspiracy theorists: I call them Conclusionists, and the pit they fall into a conclusive evidence trap.

That’s what happened with Sandeep.

“Of course I don’t think so!” I replied. “How can you say such a … thing?” I asked, working on resisting my own adjectival overpouring. He looked at me with candid disbelief. How could I be so naïve? The web is full of information about it – he said. And when I asked him to give me an example, he explained: “Of course it is not in the usual places. You need to know where to look”.

Oh my God. One tends to imagine Conclusionists as showing some exterior signs of dimwittedness. But there he was, a perfectly nice, bright-looking guy, splattering such shocking bullshit. As he excused himself to the men’s room, I tried to collect myself. But failed miserably. “So Sandeep” I asked him as he came back, even before he could regain his seat “Who killed JFK? And what about those moon landings? And the Illuminati? It’s all down to Queen Elisabeth, eh?” I deserved a sonorous expletive. But Sandeep was a gentleman, and perhaps he had regretted his own condescension over his micturating interval. “I see your point” he smiled “I’m not saying that everything you find on internet is true. But…” At which point I grabbed the two seconds void and, after mumbling some sort of apology myself, I cleared the air with a liberating “Anyway…” followed by a question about salaries, as if the whole interlude had never happened. The conversation resumed its cordial tone and carried on for a while, until it was time to go. We departed with the inevitable “Let’s keep in touch”. I never heard from him since.

They arrived in the morning, bright and early. The dishwasher had been acting strangely, so I had finally called in the engineers to figure out what was going on. I like to fix these things myself around the house, but this time, after fiddling in vain for a few days, I had given up.

“‘morning, Sir – how can we help?” Doug, the senior of the duo, had the reassuring air of the expert who has seen it all.

“Well, this is what’s happening” I started, hopeful but sceptical that Doug would immediately find an obvious explanation. “The washing cycle does not end properly. As you can see, it stops in the middle, with water still lying at the bottom. It’s not the filter or anything like that” I added, making it clear that I knew my stuff. “Sometimes, after I open and close the door a couple of times, it restarts and goes on to the end. But other times, like today, it just stops”.

“Let’s take a look” said Doug, and with a nod and a whisper instructed his younger mate Trevor to check under the sink. At this point I left, one because the children had woken up and two because watching Trevor puffing and laying his giant tattooed belly on the cold marble floor was a bit too much so early in the morning. “Call me if you need me” I said. But I had hardly greeted the kids that Doug called me back. “Here it is, Sir” – the dishwasher was working again. “It was the connection to the water drain. It is shared with the washing machine and sometimes it can be a bit too much, you know. Anyway, we’ve changed it around and it should not happen again. But remember never to use the dishwasher and the washing machine at the same time”.

“Ow…kay” I said, trying to conceal my puzzlement and following Doug’s invitation to look under the sink at the result of Trevor’s manipulation. I couldn’t see any difference – and I had never used the two machines simultaneously. “Are you sure?” I wanted to ask, but I refrained – Doug looked very sure, and ready to leave. “Thank you very much” was all I could say. “Pleasure, Sir” said Doug, “it should be alright but we’re here if you need us. Have a good day”.

Alas, the little hope I had for a quick solution soon faded away. The dishwasher finished the cycle that Doug and Trevor had managed to restart, but the next one flopped again in the middle, as I found out the following morning. A little door banging helped it to the end, and so it did in next few days. But the whole process soon became increasingly irritating: sometimes everything worked fine, sometimes the machine stopped and restarted by itself as I entered the kitchen, and some other times I had to keep banging the door. A week later I called back.

“Sorry guys” I apologised on the phone as I explained that their fixing wasn’t working. “No worries, Sir. We’ll be there tomorrow early in the morning”. So that evening I started a new cycle, with the intent of showing them the result in the morning and creating the ideal conditions for a new assessment.

I got out of bed as they rang the bell. They came in and we walked to the kitchen. One, two, three: I opened the dishwasher door, ready to show them the usual stagnant pool of water. Et voilà: no water. This time the cycle had ended properly. “No problem at all, Sir” said Doug, helping to alleviate my evident embarrassment. “We’ll put it down as ‘Intermittent Malfunctioning'”.

As they left with what I couldn’t help interpreting as a wry smile of amusement, I started contemplating my life with an erratically faulty dishwasher. Sure enough, the stop and go resumed. But what was the point of calling them again? So I kept going for a while, banging and cursing. Until one day it all came to an end. No banging, no lights, nothing. The machine was completely dead, and an increasingly smelly sludge at the bottom left me no alternative to calling Doug once again, with a view to arranging for a replacement.

This time Doug came alone, and after a few fearful moments in which I was dreading a new mysterious restart, he declared death himself. He took away the wooden bar under the dishwasher and started fiddling with its feet, exploring ways to slide it out of its casing. I left him again, and again he soon called me back. “Here, Sir” – the dishwasher had come back to life. To my befuddlement and consternation, Doug offered a new explanation: “You see, Sir, it all has to do with the alignment of the feet. They have laid the machine on MDF – that’s not the correct way, they should have used a harder material. With time, the feet have sunk a bit into the wood, enough to misalign the door closing. That’s why banging works sometimes. I have now raised the feet a bit so it’s all back in line. If this doesn’t work, the next thing is to replace the door, but I will not do it myself – I tried it once, but the hinges snapped back and I almost lost my finger. Anyway, I don’t think it will be necessary. I believe I figured it out – it’s amazing how one keeps learning after all these years”.

Oh well. I didn’t know what to make of Doug’s new theory, but he had managed to raise my hopes a bit. Once again, I would have the evidence in the morning. But later in the day I received a phone call. It was an electrician, who explained that he had been instructed by Doug to look at the dishwasher’s plug and asked whether he could come in the afternoon for a check. I was confused – Doug had said nothing to me about the plug. But why not? The whole thing was starting to reveal an amusing side.

As the young electrician came in, I gave him an abridged version of the saga. He nodded, quite uninterested, and set out to slide out the dishwasher to reach for the plug, which he had figured out was right behind it. After a few minutes he called me back. “Here, have a look” he said, with a quiet smile. The plug was stuck to the rear of the dishwasher, its plastic back partially melted and fused into it:

The mystery was finally and completely solved. And, as in the best detective stories, the explanation was simple and totally unexpected. The plug, stuck to the back, would intermittently lose contact with the socket due to the dishwasher’s vibration in mid cycle. That’s why door banging helped – it restored contact, as sometimes did just walking back into the kitchen, as floor vibration was enough to produce the same effect. All the electrician had to do was to move the socket to the side panel and reinsert the plug there. A dishwasher that was about to be chucked away is now in perfect shape and flawlessly performing its wonders.

So much for Doug’s theories. He had first tried a routine explanation – one that would probably fit most similar cases – but received disconfirming evidence from me. He then got confirming evidence from his own observation – a treacherous occurrence in many circumstances. Then, when a new piece of disconfirming evidence arrived, he built a new theory around it that seemed to fit the facts. This was as wrong as the first – and even more so, as it lacked generality and was created on the spot.

To his credit, however, Doug was crucial to finding the truth. I don’t know why he didn’t tell me about the plug – maybe it was late lateral thinking, or maybe he had it in mind but didn’t want to spoil his new-fangled theory – or simply, with no Trevor around, he didn’t feel like going through the motions of sliding the machine out.

Be as it may, Doug was a true scientist. The search for the truth proceeds neither by deduction nor by induction but – in Charles Sanders Peirce’s somewhat awkward phrasing – abduction. We test hypotheses to produce explanations and select those that provide the best explanation of the observed evidence. The key to the process is to be open to revising and possibly rejecting any explanation in the light of the observed evidence. But a true scientist goes further: he actively looks for evidence that would reject his best theory and only stops when he finds conclusive evidence. In our dishwasher tale – a true story – the fused plastic plug was a Smoking Gun: evidence that conclusively explained the dishwasher’s strange behaviour. Hence we say it was the cause of such behaviour. I sent the picture to Doug’s phone but got no reply – I can’t remember, but perhaps, unlike his owner, the phone is not a smart one.

Investment risk is the probability of a substantial and permanent loss of capital. We buy a stock at 100 expecting to earn a return, consisting of appreciation and possibly a stream of dividends. But our expectation may be disappointed: the price may go down rather than up and we may decide to sell the stock at a loss, either because we need the money or because we come to realise, rightly or wrongly, that we made a mistake and the stock will never reach our expected level.

How does investment risk relate to volatility – the standard deviation of past returns, measuring the extent to which returns have been fluctuating and vibrating around their mean? Clearly, we prefer appreciation to be as quick and smooth as possible. If our expected price level is, say, 150, we would like the stock to reach the target in a straight line rather than through a tortuous rollercoaster. On the other hand, if we are confident that the price will get there eventually, we – unimpressionable grownups – may well endure the volatility. In fact, if on its way to 150 the price dropped to 70 it would create an inviting opportunity to buy more.

Volatility increases investment risk only insofar as it manages to undermine our confidence. We might have rightly believed that Amazon was a great investment at 85 dollars in November 1999, but by the time it reached 6 two years later our conviction would have been brutally battered. Was there any indication at the time that the stock could have had such a precipitous drop? Sure, the price had been gyrating wildly until then, up 21% in November, down 12% in October, up 29% in September and 24% August, down 20% in July, and so on. The standard deviation of monthly returns since the IPO had been 33%, compared to 5% for the S&P500, suggesting that further and possibly more extreme gyrations were to be expected. But to a confident investor that only meant: tighten your seatbelt and enjoy the ride. A 93% nosedive, however, was something else – more than enough to break the steeliest nerves and crush the most assured resolve. ‘I must be wrong, I’m out of here’ is an all too human reaction in such circumstances.

Therefore, while volatility may well contribute to raise investment risk, it is not the same as investment risk. It is only when – rightly or wrongly – conviction is overwhelmed by doubt and poise surrenders to anxiety that investment risk bears its bitter fruit.

Amazon is a dramatic example, but this is true in general. Every investment is made in the expectation of making a return, together with a more or less conscious and explicit awareness that it may turn out to be a flop. Every investor knows this, in practice. So why do many of them ignore it in theory and keep using financial models built on the axiom that volatility equals investment risk? As we have seen, the reason is the intellectual dominance of the Efficient Market Theory.

So the next question is: Why is it that, according to the EMT, investment risk coincides with volatility? The answer is as simple as it is unappreciated. Let’s see.

If the EMT could be summarised in one sentence, it would be: The market price is right. Prices are always where they should be. Amazon at 85, 6 or 1000 dollars. The Nasdaq at 5000, 1400 or 6400. At each point in time, prices incorporate all available information about expected profits, returns and discount rates. Prices are never too high or too low, except with hindsight. Therefore, an investor who buys a stock at 100 because he thinks it is worth 150 is fooling himself. Nobody can beat the market. If the market is pricing the stock at 100, then that’s what it’s worth. The price will change if and only if new information – unknown and unknowable beforehand and therefore not yet incorporated into the current price – prompts the market to revise its valuation. As this was true in the past as it is true in the present and will be true in the future, past price changes must also have been caused by no other reason than the arrival of information that was new at the time and unknown until then. Thus all price changes are unknowable and, by definition, unexpected. And since price changes are the largest components of returns – the other being dividends, which can typically be anticipated to some extent – we must conclude that past returns are largely unexpected. At this point there is only one last step: to identify risk with the unexpected. If we define investment risk as anything that could happen to the stock price that is not already incorporated into its current level, then the volatility of past returns can be taken as its accurate measure.

Identifying investment risk with volatility presupposes market efficiency. This is part of what Eugene Fama calls the joint hypothesis problem. To be an active investor, thus rejecting the EMT in practice, while at the same time using financial models based on the identification of investment risk with volatility, thus assuming the EMT in theory, is a glaring but largely unnoticed inconsistency.

So the next question is: what is it that practitioners know and makes them behave as active investors, and EMT academics ignore and leads them to declare active investment an impossible waste of time and to advocate passive investment?

Again, the answer is simple but out of sight. In a nutshell: Practitioners know by ample experience that investors have different priors. EMT academics assume, by theoretical convenience, that investors have common priors.

Different priors is the overarching theme of the entire Bayes blog. People can and do reach different conclusions based on the same evidence because they interpret evidence based on different prior beliefs. This is blatantly obvious everywhere, including financial markets, where, based on the same information, some investors love Amazon and some other short it. In the hyperuranian realm of the EMT, on the other hand, investors have common priors and therefore, when faced with common knowledge, cannot but reach the same conclusion. As Robert Aumann famously demonstrated, they cannot agree to disagree. This is why, in EMT parlance, prices reflect all available information.

Take the assumption away and the whole EMT edifice comes tumbling down. This is what Paul Samuelson was referring to in the final paragraphs on the Fluctuate and Vibrate papers. More explicitly, here is how Jonathan Ingersoll put it in his magisterial Theory of Financial Decision Making, immediately after ‘proving’ the EMT:

In fact, the entire “common knowledge” assumption is “hidden” in the presumption that investors have a common prior. If investors did not have a common prior, then their expectations conditional on the public information would not necessarily be the same. In other words, the public information would properly also be subscripted as φk – not because the information differs across investors, but because its interpretation does.

In this case the proof breaks down. (p. 81).

Interestingly, on a personal note, I first made the above quotation in my D.Phil. thesis (p. 132). A nice circle back to the origin of my intellectual journey.

As he wrote his ‘Challenge to Judgment’ on the first issue of the Journal of Portfolio Management in 1974, Paul Samuelson expected ‘the world of practical operators’ and ‘the new world of academics’ – which at the time looked to him ‘still light-years apart’ – to show some degree of convergence in the future.

On the face of it, he was right. The JPM recently celebrated its 40th anniversary. The Financial Analysts Journal, started with the same bridging intent 30 years earlier under Ben Graham’s auspices, is alive and well on its 73rd Volume. Dozens other periodicals have joined in the effort and hundreds of books and manuals have been written, sharing the purpose of promoting and developing a common language connecting the practice and the theory of investing.

But, while presuming and pretending to understand each other, the two worlds are still largely immersed in a sea of miscommunication. At the base of the Babel there are two divergent perspectives on the relationship between risk and return. Everybody understands return. You buy a stock at 100 and the price goes up to 110 – that’s a 10% return. But this is ex post. What was your expected return before you bought? And what risk did you assume? The practical operator does not have precise answers to these questions. I looked at the company – he would say – studied its business, read its balance sheet, talked to the managers, did my discounted cash flow valuation and concluded that the company was worth more than 100 per share. So I expected to earn a good return over time, roughly equal to the gap between my intrinsic value estimate and the purchase price. As for risk, I knew my valuation could be wrong – the company might be worth less than I thought. And even if I was right at the time of purchase, the company and my investment might have taken wrong turns in myriads different ways, causing me to lose some or all of my money.

Is that it? – says the academic – is that all you can say? Of course not – replies the operator – I could elaborate. But I couldn’t do it any better than Ben Graham: read his books and you’ll get all the answers.

But the academic would have none of it. As Eugene Fama recalls: ‘Without being there one can’t imagine what finance was like before formal asset pricing models. For example, at Chicago and elsewhere, investment courses were about security analysis: how to pick undervalued stocks’. (My Life in Finance, p. 14). Go figure. Typically confusing science with precision, the academic is not satisfied until he can squeeze concepts into formulas and insights into numbers. I don’t know what to do with Graham’s rhetoric – he says – I need measurement. So let me repeat my questions: what was your expected return exactly? How did you quantify your risk?

Give me a break – says the defiant operator – risk is much too complex to be reduced to a number. As for my expected return, I told you it is the gap between value and price, but I am under no illusion that I know it exactly. All I know is that the gap is large enough and I am prepared to wait until it closes.

Tut-tut – Fama shakes his head – Listen to me, you waffly retrograde. I will teach you the CAPM. ‘The CAPM provides the first precise definition of risk and how it drives expected return, until then vague and sloppy concepts’. (p. 15).

The operator listens attentively and in the end says: Sorry, I think the CAPM is wrong. First, you measure risk as the standard deviation of past returns. You do it because it gives you a number, but I think it makes little sense. Second, you say the higher the risk the higher is the expected return. That makes even less sense. My idea of risk is that the more there is the more uncertain I am about my expected return. In my view, the relationship between risk and expected return is, if anything, negative. So thank you for the lecture, but I stick with Graham. As Keynes did not say (again!): It is better to be vaguely right than precisely wrong.

Writing ten years after Samuelson’s piece, Warren Buffett well expressed the chasm between academics and practical operators: ‘Our Graham & Dodd investors, needless to say, do not discuss beta, the capital asset pricing model or covariance in returns among securities. These are not subjects of any interest to them. In fact, most of them would have difficulty defining those terms. The investors simply focus on two variables: price and value’. (Buffett, Superinvestors, p. 7).

But operators are rarely so blunt. Such is the intellectual authority of the Efficient Market Theory that the identification of risk with the standard deviation of returns – a.k.a. volatility – and the implication that more risk means higher returns are taken for granted and unthinkingly applied to all sorts of financial models. Hilariously, these include the same valuations that investment practitioners employ to justify their stock selection – an activity that makes sense only if one rejects the EMT! It is pure schizophrenia: investors unlearn at work what they learned at school, while at the same time continuing to use many of the constructs of the rejected theory and failing to notice the inconsistency.

But here is the biggest irony: after teaching it for forty years – twenty after Buffett’s piece – Fama finally got it out of his system: ‘The attraction of the CAPM is that it offers powerful and intuitively pleasing predictions about how to measure risk and the relation between expected return and risk. Unfortunately, the empirical record of the model is poor – poor enough to invalidate the way it is used in applications (Fama and French, JEP 2004). Hallelujah. Never mind that in the meantime the finance world – academics and practitioners – had amassed a colossal quantity of such applications and drawn an immeasurable variety of invalid conclusions. But what is truly mindboggling is that, in spite of it all, the CAPM is still regularly taught and widely applied. It is hard to disagree with Pablo Fernandez – a valiant academic whose work brings much needed clarity amidst the finance Babel – when he calls this state of affairs unethical:

If, for any reason, a person teaches that Beta and CAPM explain something and he knows that they do not explain anything, such a person is lying. To lie is not ethical. If the person “believes” that Beta and CAPM explain something, his “belief” is due to ignorance (he has not studied enough, he has not done enough calculations, he just repeats what he heard to others…). For a professor, it is not ethical to teach about a subject that he does not know enough about.

Two books that I think are particularly effective in helping operators move from practical unlearning – erratic, undigested and incoherent – to proper intellectual unlearning of the concept of risk embedded in the EMT and its derivations are David Dreman’s Contrarian Investment Strategies: the Next Generation (particularly Chapter 14: What is Risk?) and Howard Marks’ The Most Important Thing (particularly Chapters 5-7 on Understanding, Recognizing and Controlling Risk).

Besides the EMT’s predominance, unlearning is necessary because, at first glance, measuring risk with the standard deviation of returns makes intuitive sense: the more prices ‘fluctuate’ and ‘vibrate’, the higher the risk. Take Amazon:

If you had invested 30,000 dollars in Amazon’s IPO in May 1997 (it came out at 18 dollars, equivalent to 1.50 dollars after three splits), after twenty years – as the stock price reached 1,000 dollars (on 2nd June this year, to be precise) – your investment would have been be worth 20 million dollars. Everybody understands return. But look at the chart – in log scale to give a graphic sense of what was going on: 1.50 went to 16 in a year (+126% in one month – June 1998) to reach 85 in November 1999. Then in less than two years – by September 2001 – it was down to 6, only to climb back to 53 at the end of 2003, down to 27 in July 2006, up to 89 in October 2007, down to 43 in November 2008 and finally up – up up up – to 1000. Who – apart from Rip van Winkle and Jeff Bezos – would have had the stomach to withstand such infernal rollercoaster?

So yes, in a broad sense, volatility carries risk. The more violent the price fluctuations, the higher is the probability that, for a variety of psychological and financial circumstances – he may get scared and give up on his conviction, or he may need to liquidate at the wrong time – an investor might experience a catastrophic loss. But how can such probability be measured? The routine, automatic answer is: the standard deviation of returns. Here is the picture:

The graph on the left is the cumulative standard deviation of monthly returns from May 1997 (allowing for an initial 12-month data accumulation) to May 2017, for Amazon and for the S&P500 index. The graph on the right shows the 12-month rolling standard deviations. The cumulative graph, which uses the maximal amount of data, shows that while the monthly standard deviation of the S&P500 has been stable at around 5%, Amazon’s standard deviation has been, after an initial peak, steadily declining ever since, although it still remains about four times that of the index (18.4% vs. 4.4%). The 12-month rolling version shows a similar gap, with Amazon’s standard deviation currently about three times that of the S&P500 (5.1% vs. 1.8%).

What does this mean? Why is it relevant? What can such information tell us about the probability that, if we buy Amazon today, we may incur a big loss in the future? A moment’s thought gives us the answer: very little. Clearly, today’s Amazon is a completely different entity compared to its early days in the ’90s. Using any data from back then to guide today’s investment decision is nothing short of mindless. Amazon today is not four times as risky as the market, as it wasn’t five times as risky in November 2008. Nor is it three times as risky, as implied by the 12-month rolling data. The obvious point is that the standard deviation of returns is a backward-looking, time-dependent and virtually meaningless number, which, contrary to the precision it pretends to convey, has only the vaguest relation to anything resembling what it purports to measure.

The same is true for the other CAPM-based, but still commonly used measure of risk: beta. Here is Amazon’s beta versus the S&P500 index, again cumulative and on a 12-month rolling basis:

Again, the cumulative graph shows that Amazon’s beta has always been high, though it has halved over time from 4 to 2. So is Amazon a high beta stock? Not according to the 12-month rolling measure, which today is 0.4 – Amazon is less risky than the market! – but has been all over the place in the past, from as high as 6.7 in 2007 to as low as -0.4 in 2009. Longer rolling measures give a similar picture. What does it mean? Again, very little. According to the CAPM, Amazon’s beta is supposed to be a constant or at least stable coefficient, measuring the stock’s sensitivity to general market movements. But in reality it is nothing of the kind: like the standard deviation of returns, beta is just an erratic, retrospective and ultimately insignificant number.

Volatility implies risk. But reducing risk to volatility is wrong, ill-conceived and in itself risky, as it inspires the second leg of the CAPM misconception: the positive relationship between risk and expected return. ‘Be brave, don’t worry about the rollercoaster – you’ll be fine in the end and you’ll get a premium. The more risk you are willing to bear, the higher the risk premium you will earn.’ Another moment’s reflection is hardly necessary to reveal the foolishness – and to commiserate the untold damage – of such misguided line of reasoning. The operator’s common sense view is correct: once risk is properly defined as the probability of a substantial and permanent loss of capital, the more risk there is the lower – not the higher – is the probability-weighted expected return. This also requires unlearning – often, alas, the hard way.

Despite Samuelson’s best wishes, then, there is far less authentic common ground between operators and academics than what is pretended – in more or less good faith – in both camps. Operators are right: there is much more to risk than volatility and beta, and actual risk earns no premium.

In the latest chapter of his life-long and eventually triumphal effort to promote index investing, John Bogle explains what lays at the foundation of his philosophy: ‘my first-hand experience in trying but failing to select winning managers’ (p. 6). In 1966, as the new 37-year old CEO of Wellington Management Company, Bogle decided to merge the firm with ‘a small equity fund manager that jumped on the Go-Go bandwagon of the late 1960s, only to fail miserably in the subsequent bear market. A great – but expensive – lesson’ (p. 7), which cost him his job.

It reminded me of another self-confessed failure, as recounted by Eugene Fama, who in his young days worked as a stock market forecaster for his economics professor, Harry Ernst: ‘Part of my job was to invent schemes to forecast the market. The schemes always worked on the data used to design them. But Harry was a good statistician, and he insisted on out-of-sample tests. My schemes invariably failed those tests’. (My Life in Finance, p. 3).

I can’t help seeing both incidents as instances of Festinger’s cognitive dissonance. It runs more or less like this: 1) I know a lot about economics and stock markets. 2) I am smart – truth be told, very smart. 3) I could use my brains to predict stock prices/select winning managers and make a lot of money. 4) I can’t. Therefore: it must be impossible. I think this goes a long way towards explaining the popularity and intuitive appeal of the Efficient Market Theory in academia.

Typical academics are keen to take these as conclusive demonstrations – derived from first principles, like Euclidean theorems – of the impossibility of market beating. But the Master knew better. At the end of ‘Fluctuate’ he wrote:

I have not here discussed where the basic probability distributions are supposed to come from. In whose minds are they ex ante? In there any ex post validation of them? Are they supposed to belong to the market as a whole? And what does that mean? Are they supposed to belong to the “representative individual”, and who is he? Are they some defensible or necessitous compromise of divergent expectation patterns? Do price quotations somehow produce a Pareto-optimal configuration of ex ante subjective probabilities? This paper has not attempted to pronounce on these interesting questions.

And at the end of ‘Vibrate’:

In summary, the present study shows (a) there is no incompatibility in principle between the so-called random-walk model and the fundamentalists’ model, and (b) there is no incompatibility in principle between behaviour of stocks’ prices that behave like random walk at the same time that there exists subsets of investors who can do systematically better than the average investors.

Then in 1974 he reiterated the point in crystal clear terms, addressed to both academics and practitioners on the first issue of the Journal of Portfolio Management:

What is at issue is not whether, as a matter of logic or brute fact, there could exist a subset of the decision makers in the market capable of doing better than the averages on a repeatable, sustainable basis. There is nothing in the mathematics of random walks or Brownian movements that (a) proves this to be impossible, or (b) postulates that it is in fact impossible. (Challenge to Judgment, p. 17, his italics).

And for the EMT zealots:

Many academic economists fall implicitly into confusion on this point. They think that the truth of the efficient market or random walk (or, more precisely, fair-martingale) hypothesis is established by logical tautology or by the same empirical certainty as the proposition that nickels sell for less than dimes.

The nearest thing to a deductive proof of a theorem suggestive of the fair-game hypothesis is that provided in my two articles on why properly anticipated speculative prices do vibrate randomly. But of course, the weasel words “properly anticipated” provide the gasoline that drives the tautology to its conclusion. (p. 19).

There goes ‘Bogle’s truth’. And the irony of it is that in his latest piece Bogle reminisces on how, as he read it at the time, ‘Dr. Samuelson’s essay … struck me like a bolt of lightning’ (p. 6). A hard, obnubilating blow indeed.

There was, nevertheless, a legitimate reason for the fulmination. Samuelson’s Challenge to Judgment was a call to practitioners:

What is interesting is the empirical fact that it is virtually impossible for academic researchers with access to the published records to identify any member of the subset with flair. This fact, though not an inevitable law, is a brute fact. The ball, as I have already noted, is in the court of those who doubt the random walk hypothesis. They can dispose of the uncomfortable brute fact in the only way that any fact is disposed of – by producing brute evidence to the contrary. (p. 19).

He was referring to Jensen (1968) and the copious subsequent literature presenting lack of evidence on identifying a consistent subset of long-term outperforming funds. What Samuelson missed, however – and still goes largely unnoticed – is that the ‘risk adjustments’ to fund and index returns used in these studies are based on definitions of risk – as volatility, beta and the like – that presume market efficiency. To his credit, Eugene Fama has always been very clear on this point, which he calls the joint hypothesis problem:

Market efficiency can only be tested in the context of an asset pricing model that specifies equilibrium expected returns. […] As a result, market efficiency per se is not testable. […] Almost all asset pricing models assume asset markets are efficient, so tests of these models are joint tests of the models and market efficiency. Asset pricing and market efficiency are forever joined at the hip. (My Life in Finance, p. 5-6).

Typically, outperforming funds are explained away, and their returns driven to statistical insignificance, by the ‘higher risk’ they are deemed to have assumed. But such risk is defined and measured according to some version of the EMT! It is – as James Tobin wryly put it – a game where you win when you lose (see Tobin’s comment to Robert Merton’s essay in this collection).

It was precisely in defiance of this game that Warren Buffett wrote his marvellous Superinvestors piece, which sits up there next to Ben Graham’s masterwork in every intelligent investor’s reading list. As in his latest shareholder letter, Buffett used the coin-flipping story, fit for humans as well as orangutans, to point out that past outperformance can be the product of chance. But then he drew attention to an important difference:

If (a) you had taken 225 million orangutans distributed roughly as the U.S. population is; if (b) 215 winners were left after 20 days; and if (c) you found that 40 came from a particular zoo in Omaha, you would be pretty sure you were on to something. So you would probably go out and ask the zookeeper about what he’s feeding them, whether they had special exercises, what books they read, and who knows what else. That is, if you found any really extraordinary concentrations of success, you might want to see if you could identify concentrations of unusual characteristics that might be causal factors. (p. 6).

Hence he proceeded to illustrate the track record of his nine Superinvestors, stressing that it was not an ex post rationalisation of past results but a validation of superior stock picking abilities that he had pre-identified ex ante.

So let’s do a thought experiment and imagine that Buffett 2007 went back 40 years to 1967 and wagered a bet: ‘I will give 82,000 dollars (about 500,000 2007 dollars in 1967 money) to any investment pro who can select five funds that will match the performance of the S&P500 index in the next ten years’. Would Buffett 1967 have taken the bet? Sure – he would have said – in fact, I got nine! And after nine years, one year prior to the end of the bet, he would have proclaimed his victory (I haven’t done the calculation on Buffett’s Tables, but I guess it’s right). Now let’s teleport Buffett 2016 to 1976. What would he have said? Would he have endorsed those funds or recommended investing in the then newly launched Vanguard S&P index fund?

Here is then why I am disoriented – and I’m sure I’m not alone – by Mr. Buffett’s current stance on index investing. To be clear: 1) I am sympathetic to his aversion to Buffett impersonators promoting mediocre and expensive hedge funds. 2) I think index funds can be the right choice for certain kinds of savers. 3) I think Jack Bogle is an earnest and honourable man. However, as a grateful and impassioned admirer of Buffett 1984, Buffett 2016 puzzles me. Like the former, the latter agrees with Paul Samuelson against ‘Bogle’s truth’: long term outperformance, while difficult and therefore uncommon – no one denies it – is possible. But while Buffett 1984 eloquently expanded on the ‘intellectual origin’ (p. 6) of such possibility, and on the ex ante characteristics of superior investors, Buffett 2016’s message is: forget about it, don’t fall for ex post performance and stick to index funds.

Notice this is not a message for the general public: it is addressed to Berkshire Hathaway’s shareholders – hardly the know-nothing savers who may be better served by basic funds. Buffett is very clear about this: buying a low-cost S&P500 index fund is his ‘regular recommendation’ (p. 24), to large and small, individual as well professional and institutional investors – noticeably including the trustees of his family estate (2013 shareholder letter, p. 20).

Great! There goes a life-long dedication to intelligent investing. You may as well throw away your copy of Security Analysis. Alternatively, you may disagree with Mr. Buffett – nobody is perfect – and hope he reconsiders his uncharacteristically unfocused analysis. From the Master who taught us how to select good stocks one would expect equivalent wisdom on how to select good funds. It is not the same thing, but there are many similarities. As in stock picking, there are many wrong things one can do in fund picking. Past performance is no guarantee of future performance. Expensive stocks as well as expensive funds deceptively draw investors’ attention. There is no reason why large stocks or large funds should do better than small ones. Don’t go with the crowd. And so on. Similarly, just like Mr. Buffett taught us how to do the right things in stock picking, he could easily impart comparable advice in fund picking.

Here is the first one that comes to mind: look at the first ten stocks in a fund and ask the fund manager why he holds them. If he makes any reference to their index weight, run away.

Look at the top holdings of Italian Equities funds (Azionari Italia) on morningstar.it. They are the same for most of them: ENI, Intesa Sanpaolo, Enel, Unicredit, Luxottica, Assicurazioni Generali, Fiat Chrysler, and so on. Why? Do most fund managers agree that these are the best and most attractive companies quoted on the Italian stock market? No. The reason is that these are the largest companies by market capitalization, and therefore the largest components of the most commonly used Italian Equities index, the FTSE MIB. The same is true for other countries and regions, as well as for sector funds: look at the composition of the relevant index and you will work out a large portion of the funds’ holdings.

To a candid layman this looks very strange. ENI may be a good company, but why should it be as much as 10% of an Italian Equities fund? Surely, a company’s size has nothing to do with how valuable it is as an investment. Aren’t there more attractive choices? And if so, shouldn’t the fund invest in them, rather than park most of the money in the larger companies?

No, is the fund manager’s answer: the fund’s objective is not simply to find attractive investments. It is to obtain over time a better return than its peers and the index. This is what drives investors’ choices, determines the fund’s success and its manager’s reward. To beat the index – says the manager – I have to face it: take it as a neutral position and vary weights around it. So if I think that ENI is fairly valued I will hold its index weight, if I think it is undervalued I will hold more, and if I think it is overvalued I will hold less. How much more or less is up to me. But if ENI is 10% of the index I would have to regard it as grossly overvalued before deciding to hold none of it in the fund. A zero weight would be a huge bet against the index, which, if it goes wrong – ENI does well and I don’t have it – would hurt the fund’s relative performance and my career.

Sorry to insist – says the outspoken layman – but shouldn’t the fund’s performance and your career be better served if you take that 10% and invest it in stocks that you think will do better than ENI? If you do the same with the other large stocks which, like ENI, you hold in the fund just because they are in the index, you may be wrong a few times, but if you are any good at stock picking – and you tell me you are, that’s why I should buy your fund – then surely you are going to do much better than the index. What am I missing?

Look sir, with all due respect – says the slightly irritated manager – let me do my job. You want the fund to outperform, and so do I. So let me decide how best to achieve that goal, if you don’t mind.

I do mind – says the cheeky layman, himself showing signs of impatience. Of course I want you to beat the index. But I want you to do it with all my money, not just some of it. The index is just a measure of the overall market value. If ENI is worth 53 billion euro and the whole Italian stock market is worth 560 billion – less than Apple, by the way – then, sure, ENI is about 10% of the market. But what does that have to do with how much I, you or anybody else should own of it? The market includes all stocks – the good, the bad and the ugly. If you are able to choose the best stocks, you should comfortably do better than the market. If you can’t, I will look somewhere else.

Oh yeah? Good luck with that – the manager has given up his professional demeanour – hasn’t anybody told you that most funds do worse than the index?

Yes, I am aware of it – says the layman – that’s why I am looking for the few funds that can do better. You’re right, if your peers do what you do, I am not surprised they can’t beat the index. But I’ll keep looking. Good bye.

Well done, sir – someone else approaches the layman – let me introduce myself: I am the indexer. You’re right, all this overweight and underweight business is a complete waste of time and money. The reality is that, sooner or later, most funds underperform the index – and they even want to get paid for it! So let me tell you what I do: in my fund, I hold the stocks in the index at exactly their neutral weight, but I charge a small fraction of the other funds’ fees. This way, my fund does better than most other funds, at a much lower cost. How does that sound?

Pretty awful, I must say – says the layman – I am looking for a fund that invests all my money in good stocks and you are proposing one that does none of that and mindlessly buys index stocks. And you call yourself an investor?

Pardon me, but you’re so naïve – says the indexer – I am telling you I do better than most, at a lower cost. What part of the message don’t you understand?

Well, it’s not true – say the layman – and proceeds to show the indexer a list of funds that have done better than the relevant index and the other funds for each category over several periods after all costs – he may be a layman but he’s done his homework.

Oh, that’s rubbish – retorts the indexer – and performs his well-rehearsed coin-tossing gig. These are just the lucky guys who happen to sit on the right tail of the return distribution for a while. Sooner or later, their performance will revert to the mean. And do you know why? Because markets are efficient. Have you heard of the Efficient Market Theory? – he asks with a smug look. There is tons of academic evidence that proves that consistent market beating is impossible.

Yes, I know the EMT – says the layman – and I think it is wrong. Beating the market is clearly difficult – if it were easy everybody could do it, hence nobody would – but it is not impossible. The numbers I just showed you prove my point, and to dismiss them as a fluke is a miserable argument, fit only for haughty academics in need of a soothing answer to a most nagging question: If you’re so smart, why aren’t you rich? Tell me something – continues the layman – what drives market efficiency? Certainly not you, or the other gentleman with his marginal tweaking. You buy any company in the index regardless of price.

Yes – says the indexer, hiding his discomfort – but we are powerful and responsible shareholders and make sure that our voice gets heard.

Give me a break – the layman laughs – companies don’t care about you. They know you have to hold their shares no matter what. You’re the epitome of an empty threat. You don’t even know or care what these companies do. You are not an investor – you’re a free rider.

Ok then – says the indexer (he knew his was a phony argument but he tried it anyway) – what’s wrong with that? If there are enough active investors busy driving prices to where they should be, my passive fund reaps the benefits, my investors pay less and everyone is happy.

You should be ashamed of yourself, you know – says the layman, ready to end his second conversation.

Aw come on now! – blurts the indexer – who’s worse: me, transparently declaring what I do and charging little for it, or the other guy, pretending to be smart, doing worse than me and charging ten times as much?

You’ve got a point there – says the layman – you’re better than him. But you’re not going to get my money either. Good bye.

As you like, it’s your money – says the indexer, before launching his departing salvo: you know, even Warren Buffett says that index investing is the smart thing to do.

I have seen that – says the layman – what was he thinking?

Yes, what was Warren Buffett thinking when in his 2016 shareholder letter he proposed (p. 24) to erect a statue to John Bogle? Let’s see.

Back in the 2005 letter, Buffett prognosticated that active managers would, in aggregate, underperform the US stock market. He was reiterating the ‘fundamental truth’ of index investing. In the latest words of its inventor and proselytiser:

Before intermediation costs are deducted, the returns earned by equity investors as a group precisely equal the returns of the stock market itself. After costs, therefore, investors earn lower-than-market returns. (p. 2)

In its most general sense, this is an obvious tautology: the aggregate return equals the market return by definition. However, ‘Bogle’s truth’ is usually intended to apply as well to mutual funds, which for US equities represent about 20% of the aggregate (see e.g. Exhibit 3 here). As such, there is no logical reason why mutual funds should necessarily perform like the market as a group, and worse than the market after costs. In fact, a layman would be justified in expecting professional investors to do better, before and after costs, compared to e.g. households. Whether mutual funds do better than the market is therefore an empirical rather than a logical matter.

The question has a long history, dating back to Jensen (1968) all the way to the latest S&P SPIVA report. Most of these studies make it particularly hard for outperformance to show up. Rather than squarely comparing fund returns to the market index, they either adjust performance for ‘risk’ (Jensen) using the now abandoned CAPM model, or slice and dice fund returns (SPIVA), box them into a variety of categories and compare them to artificial sub-indices. As a result, the commonly held view – reflected in Buffett’s 2005 prediction – is that ‘most funds underperform the market’. From this, the allure of index investing is a small logical step and a seemingly impregnable conclusion. All you need to say is, as Buffett puts it (p. 24):

There are, of course, some skilled individuals who are highly likely to out-perform the S&P over long stretches. In my lifetime, though, I’ve identified – early on – only ten or so professionals that I expected would accomplish this feat.

There are no doubt many hundreds of people – perhaps thousands – whom I have never met and whose abilities would equal those of the people I’ve identified. The job, after all, is not impossible. The problem simply is that the great majority of managers who attempt to over-perform will fail. The probability is also very high that the person soliciting your funds will not be the exception who does well.

Further complicating the quest for worthy managers – says Buffett – is the fact that outperformance may well be the result of luck over short periods, and that It typically attracts a torrent of money, which the manager gladly accepts to his own benefit, thus making future returns more difficult to sustain.

The bottom line: When trillions of dollars are managed by Wall Streeters charging high fees, it will usually be the managers who reap outsized profits, not the clients. Both large and small investors should stick with low-cost index funds.

It was on this basis that Buffett followed his 2005 prophesy by offering a bet to any investment professional able to select at least five hedge funds that would match the performance of a Vanguard S&P500 index fund over the subsequent ten years. He called for hedge funds, which represent an even smaller portion of the US equity investor universe, as he considers them as the most strident example of divergence between bold return promises – reflected in hefty fees – and actual results. Most hedge funds do not set beating the S&P500 as their stated objective, preferring instead to target high returns independent of market conditions. But Buffett’s call was right: what’s the point of charging high fees if you can’t deliver more than index returns? At the same time, presumably he would not have objected to betting against long-only active funds explicitly managed to achieve S&P500 outperformance.

What followed – said Buffett – was the sound of silence. This is indeed surprising. Hedge fund managers’ objectives may be fuzzier, but if you manage a long-only US equity fund with a mandate to outperform the S&P500 and you genuinely believe you can do it, what better promotional opportunity is there than to bet against Warren Buffett and win?

Be as it may, only one manager took up the challenge. And – bless him – he did not choose five long-only funds, nor five hedge funds, but five funds of hedge funds: he picked five funds that picked more than 100 hedge funds that picked thousands of stocks. Nothing wrong with that, in principle. Presumably, each of the five funds of funds managers believed he could select a portfolio of hedge funds that, at least on average, would do so much better than the S&P500 that, despite the double fee layer, it would itself end up well ahead of the index. They were wrong, very wrong (p. 22). Over the nine years from 2008 to 2016, the S&P500 returned 85.4% (7.1% per annum). Only fund of funds C got somewhat close, with a return of 62.8% (5.6% per annum). The other four funds returned, in order: 28.3%, 8.7%, 7.5% and 2.9% (that is 2.8%, 0.9%, 0.8% and 0.3% per annum).

Result: Buffett’s valiant and solitary challenger, Mr. Ted Seides, co-manager, at the time, of Protégé Partners, played a very bad hand and made a fool of himself. But Buffett was lucky: he set out to prove ‘Bogle’s truth’ and observe index-like returns before fees, turning into underperformance after fees, but what he got was abysmal returns. Except perhaps for fund C, the gaping hole between the funds and the S&P500 had very little to do with fees. Buffett estimated that about 60% of all gains achieved by the five funds of funds went into the two fee layers. But even if fund D, returning a whopping 0.3% per year, had charged nothing, to select hedge funds that charged nothing, it would still have ended up well below the index. Same for funds A and E and, likely, for fund B.

To recap: when applied to mutual and hedge funds, ‘Bogle’s truth’ is not a logical necessity – as it is often portrayed to be – but is an empirical statement. Performance studies make it hard for outperformance to emerge, but beating the index in the long run is certainly no easy task, even for professional investors. Fees make it even harder – the higher the fees, the harder the task. However, while difficult to achieve and therefore rare to observe, long-term outperformance is not impossible – Buffett is the first to acknowledge it: he’s a living proof!

Why is it then that he interpreted his bet win against Seides as evidence of ‘Bogle’s truth’? Imagine he had called for five value stocks and got five duds. Would he have interpreted this as evidence of the impossibility of value investing? What’s the difference between picking stocks and picking funds? Why does Buffett consider the former a difficult but valiant endeavour while the latter an impossible waste of time?

One of the beauties of maths is that it is the same in every language. So you don’t need to know Italian to read the table on the second page of this article on this week’s Milano Finanza.

The Made in Italy Fund started in May last year and is up 43% since then.

Here are the main points of the article:

The Italian stock market is dominated by the largest 40 stocks included in the FTSE MIB index. The FTSE MIB and the FTSE Italia All Shares indices are virtually overlapping (first graph on page 1).

2/3 of the Italian market is concentrated in 4 sectors.

Small Caps – companies with a market cap of less than 1 billion euro – are 3/4 of the 320 quoted names, but represent only 6% of the value of the market.

Small Caps as a whole have underperformed Large Caps (second graph).

But quality Small Caps – those included in the Star segment of the market – have outclassed the MIB index (third graph).

However, the Star index is itself concentrated (table on page 3): the top 11 stocks in the index with a market cap above 1 billion (not 12: Yoox is no longer there) represent more than 60% of the index value (a company needs to be below 1 billion to get into the Star segment, but it is not necessarily taken out when it goes above).

Therefore, to invest in Italian Small Caps you need to know what you’re doing: you can’t just buy a Mid/Small Cap ETF – which is what a lot of people did in the first quarter of this year, after the launch of PIR accounts (similar to UK ISAs), taking the Lyxor FTSE Italia Mid Cap ETF from 42 to 469 million.

To this I would add: you can’t just buy a fund tracking the Star index either (there are a couple): to own a stock just because it is part of an index makes no sense – more on this in the next post.